3.10 \(\int \frac {(a x^2+b x^3+c x^4)^2}{x^2} \, dx\)

Optimal. Leaf size=54 \[ \frac {a^2 x^3}{3}+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \]

[Out]

1/3*a^2*x^3+1/2*a*b*x^4+1/5*(2*a*c+b^2)*x^5+1/3*b*c*x^6+1/7*c^2*x^7

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Rubi [A]  time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1585, 698} \[ \frac {a^2 x^3}{3}+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^2/x^2,x]

[Out]

(a^2*x^3)/3 + (a*b*x^4)/2 + ((b^2 + 2*a*c)*x^5)/5 + (b*c*x^6)/3 + (c^2*x^7)/7

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rubi steps

\begin {align*} \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx &=\int x^2 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^2+2 a b x^3+\left (b^2+2 a c\right ) x^4+2 b c x^5+c^2 x^6\right ) \, dx\\ &=\frac {a^2 x^3}{3}+\frac {1}{2} a b x^4+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 54, normalized size = 1.00 \[ \frac {a^2 x^3}{3}+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^3 + c*x^4)^2/x^2,x]

[Out]

(a^2*x^3)/3 + (a*b*x^4)/2 + ((b^2 + 2*a*c)*x^5)/5 + (b*c*x^6)/3 + (c^2*x^7)/7

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fricas [A]  time = 0.58, size = 44, normalized size = 0.81 \[ \frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^2/x^2,x, algorithm="fricas")

[Out]

1/7*c^2*x^7 + 1/3*b*c*x^6 + 1/2*a*b*x^4 + 1/5*(b^2 + 2*a*c)*x^5 + 1/3*a^2*x^3

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giac [A]  time = 0.48, size = 46, normalized size = 0.85 \[ \frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{5} \, b^{2} x^{5} + \frac {2}{5} \, a c x^{5} + \frac {1}{2} \, a b x^{4} + \frac {1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^2/x^2,x, algorithm="giac")

[Out]

1/7*c^2*x^7 + 1/3*b*c*x^6 + 1/5*b^2*x^5 + 2/5*a*c*x^5 + 1/2*a*b*x^4 + 1/3*a^2*x^3

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maple [A]  time = 0.00, size = 45, normalized size = 0.83 \[ \frac {c^{2} x^{7}}{7}+\frac {b c \,x^{6}}{3}+\frac {a b \,x^{4}}{2}+\frac {a^{2} x^{3}}{3}+\frac {\left (2 a c +b^{2}\right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^3+a*x^2)^2/x^2,x)

[Out]

1/3*a^2*x^3+1/2*a*b*x^4+1/5*(2*a*c+b^2)*x^5+1/3*b*c*x^6+1/7*c^2*x^7

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maxima [A]  time = 0.42, size = 44, normalized size = 0.81 \[ \frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^2/x^2,x, algorithm="maxima")

[Out]

1/7*c^2*x^7 + 1/3*b*c*x^6 + 1/2*a*b*x^4 + 1/5*(b^2 + 2*a*c)*x^5 + 1/3*a^2*x^3

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mupad [B]  time = 0.02, size = 45, normalized size = 0.83 \[ x^5\,\left (\frac {b^2}{5}+\frac {2\,a\,c}{5}\right )+\frac {a^2\,x^3}{3}+\frac {c^2\,x^7}{7}+\frac {a\,b\,x^4}{2}+\frac {b\,c\,x^6}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2 + b*x^3 + c*x^4)^2/x^2,x)

[Out]

x^5*((2*a*c)/5 + b^2/5) + (a^2*x^3)/3 + (c^2*x^7)/7 + (a*b*x^4)/2 + (b*c*x^6)/3

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sympy [A]  time = 0.08, size = 48, normalized size = 0.89 \[ \frac {a^{2} x^{3}}{3} + \frac {a b x^{4}}{2} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{7}}{7} + x^{5} \left (\frac {2 a c}{5} + \frac {b^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**3+a*x**2)**2/x**2,x)

[Out]

a**2*x**3/3 + a*b*x**4/2 + b*c*x**6/3 + c**2*x**7/7 + x**5*(2*a*c/5 + b**2/5)

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